Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

learn more… | top users | synonyms (1)

0
votes
1answer
42 views
+50

What is the differential of the quotient map?

We can view the projective space $P(\mathbb R^n)$ as the quotient of $S^n/\sim$ where $x \sim y$ if and only if $x = -y$. The quotient map $q: S^n \to P(\mathbb R^n)$ is the map $x \mapsto [x]$ ...
2
votes
1answer
20 views

Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have three linearly independent vector fields ...
6
votes
0answers
116 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
0
votes
0answers
7 views

open set in tangentspace induces open set in tangentbundle (for homogeneous spaces)

Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives. Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ ...
1
vote
1answer
31 views

Vector Fields on $\mathbf R^2$ [on hold]

Let $X : \mathbf R^2 \to \mathbf R^2$ be a no-zero smooth vector field. I want to show (without background about vector bundles or manifolds, just if possible differentiable calculus in $\mathbf R^n$) ...
3
votes
0answers
33 views

Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = ...
0
votes
0answers
12 views

Definition of locally trivial fibration

the following concerns Prop. 2.2 b) of Perelman's paper from ICM 94. What is a locally trivial fibration in this case? For instance, consider the distance function associated to the unit circle in ...
0
votes
1answer
14 views

Functions with nonnegative laplacian on Rimannian manifold.

I am doing the exercises in Do Carmo's "Riemannian Geometry". I am stuck on exercise 3.12 which states the following: Let $M$ be a compact orientalbe Riemannian manifold which is also connected. Let ...
0
votes
0answers
12 views

Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
2
votes
1answer
88 views

Pushforward of a vector

The push forward of vectors allows to transform the components of a vector $X$ to be pushed along a map $h:\mathcal{M}\to\mathcal{N}$ between the manifolds $\mathcal{M}$ and $\mathcal{N}$. This is ...
1
vote
2answers
34 views

Laplacian of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M f$ in terms of $\Delta^{S}f$ ? ((i.e the relation between ...
5
votes
2answers
35 views

Killing fields on product metrics

Let $(M_i,g_i)$ be Riemannian manifolds, $i=1,2$. (Save Euclidiean factors) Is it true that a Killing field $Z$ on $(M_1\times M_2,g_1\times g_2)$ will split as a sum of Killing fields $Z=X+Y$, where ...
0
votes
0answers
25 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
0
votes
0answers
21 views

Isothermal coordinates [duplicate]

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
2
votes
1answer
18 views

Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation?

I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where ...
2
votes
0answers
19 views

Calculating the differential of the quotient map using curves

We can view the projective space $P(\mathbb R^n)$ as the quotient of $S^n/\sim$ where $x \sim y$ if and only if $x = -y$. The quotient map $F: S^n \to P(\mathbb R^n)$ is the map $x \mapsto [x]$ ...
3
votes
1answer
180 views

How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
2
votes
0answers
29 views

Self-commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set ...
2
votes
0answers
22 views

Elementary properties of gradient systems

Consider $x_0\in\mathbb{R}^n$ and a $C^{1,1}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ (that is, a differentiable function whose gradient is Lipschitz function). Consider the system $$ ...
0
votes
1answer
19 views

The lenght of rectifiable curve in $\mathbb{R}^n \setminus B[0,r]$ that connects antipodes points.

This question is from my homework, here it goes: Let $\gamma \colon [a,b] \to \mathbb{R}^n \setminus B[0,r]$ be a rectifiable curve such that $\gamma(a)=-\gamma(b)$. Using euclidean norm prove that ...
1
vote
0answers
10 views

Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
3
votes
1answer
33 views

References on the moduli space of flat connections as a symplectic reduction

In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be ...
-1
votes
1answer
38 views

Why the topological dimension of C is 2?

From what I know, the topological dimension of a set has to do with open sets covering it, homeomorphic to R^{n}. Then we can cover C with balls, for instance,of ...
0
votes
1answer
20 views

Is an isometry between compact boundaryless embedded surfaces necessarily a rigid motion of $\mathbb{R^3}$?

A friend and I were discussing this and related questions as part of pre-exam revision, and we don't know how to answer this particular question (could not think of a proof or counterexample). Any ...
0
votes
1answer
36 views

Given a $1$-form $\omega$ on $\Bbb R^n$, is there a connection whose torsion is $T(X,Y)=\omega(X)Y-\omega(Y)X$?

Consider $(R^n, g_0 )$, where $g_0$ is the Euclidean metric, and a differential $1$-form $\omega$ on $R^n$. Can this differential form define a connection on $M=R^n$ such that its torsion is ...
1
vote
0answers
35 views

Riemannian metric as an operator

In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - ...
1
vote
1answer
19 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
0
votes
0answers
15 views

Lie bracket and local group

How to prove this identity? X and Y and smooth vector field on smooth manifold M; $\theta_t$ is the local group (one-parameter group of diffeomorphism) of Y. ...
1
vote
2answers
32 views

Local defining functions on real hypersurfaces.

I'm currently reading Real Submanifolds in Complex Space and their Mappings by Baouedi, Ebenfelt and Rothschild. I'm currently stumped by what the author's claim to be an easy check (really blowing ...
0
votes
1answer
24 views

If F is a diffeomorphism and $F_*$ preserves dot products, then F is an isometry.

Exercise from O'neill's book ELEMENTARY DIFFERENTIAL GEOMETRY (p.121) $If \quad F:R^3\to R^3\quad is \quad a\quad diffeomorphism\quad such \quad that \quad (its\quad tangent\quad map)\quad F_*\quad ...
0
votes
0answers
12 views

Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element ...
4
votes
2answers
53 views

Dimensions of immersions vs embeddings

Let's say that you have a manifold which you know can be immersed in $\mathbb{R}^n$. Is there a $k$ such that you can say, for sure, that the manifold is embedded in $\mathbb{R}^{n+k}$? I imagine that ...
1
vote
3answers
104 views

What are interesting corollaries of a manifold being parallellizable?

This is a heavily edited (in fact, a complete rewrite) of a question I asked badly a few days ago. I am editing as opposed to asking a new question as there are already several relevant answers. I ...
1
vote
1answer
13 views

Specific values of paramaters for which curve is closed

In my study of curves, I encountered this family of parametrized curves in $ \mathbb{R}^2 $ $ \cosh(y)=-A\cos(x)+B $ for real parameters A and B such that $ 0 < |A| < 1 $ My problem is to ...
2
votes
0answers
22 views

Determining whether a Lie group contains more than one conjugacy class of subgroups of a particular isomorphism type

Suppose I have a Lie group $G$. How can one determine whether there is more than one conjugacy class in $G$ of subgroups isomorphic to a given Lie subgroup $H$? Put another way: Fix a Lie ...
6
votes
1answer
78 views

Area form for $M^2 \subseteq \Bbb R^4$

We know that in general, given a orientable hypersurface $M^{n-1} \subseteq \Bbb R^n$, the volume form on $M$ is given by $$dM = \sum_{i=1}^n(-1)^{i-1}n_i\,dx^1 \wedge\cdots\wedge \widehat{dx^i}\wedge ...
3
votes
1answer
32 views

Compute the characteristic strip and conoid solution of a geometric PDE

This is a problem from Fritz John's Partial Differential Equations, which I'm working through for self-study. Given a family of spheres of radius $1$ with centers in the $xy$-plane ...
3
votes
1answer
52 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha ...
2
votes
0answers
33 views

Infinite surface area

I am reading an article (reference: http://www.jstor.org/stable/1971139?seq=1#page_scan_tab_contents), and in the proof of the main theorem, the author states that "it is a fact that complete, ...
0
votes
3answers
37 views

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$ Since evolute of a curve is the envelope of the normals of that curve.I ...
0
votes
1answer
20 views

Construct vector field along a curve

Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along ...
3
votes
1answer
1k views

Directional derivative along a curve (Covariant derivatives)

I am having a hard time understanding covariant derivatives. My main problem is working with concrete example. So I was wondering if anybody could help me with explaining it by using simple example. ...
0
votes
2answers
46 views

Gradient vector proof

Question: Prove that a normal vector to the surface $f(x,y) = \sqrt {xy}$ at any point on the surface is perpendicular to the line joining the point to the origin. I am not sure how to do this. ...
1
vote
2answers
43 views

Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?

Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an ...
1
vote
2answers
25 views

Compact Lie subgroup of $GL_n(\mathbb{R})$

Let $K\leq GL_n(\mathbb{R})$ be a compact Lie subgroup. I need to prove that $K$ is a conjugate of a subgroup of $O(n)$. The hint is to use the Haar measure, but I really don't see how to do this.
1
vote
1answer
16 views

How to find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$?

How do I find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$? I have found that the tangent plane is $z-16x-2y=95$ but I don't know how to find the normal line. The answer is: $$\frac{2 ...
0
votes
1answer
15 views

Equiareal mapping between surfaces

I have a surface parametrized with $(u,v)$ with determinant of the first fundamenthal form $\Delta =EG-F^2=\cosh \sigma +v^2\kappa^2.$ Now I'm looking for reparametrization whose Jacobian $J$ will ...
0
votes
0answers
11 views

Compact-Open Topology for Space of C^{r} -sections

Given a smooth fibre bundle $\pi: X \rightarrow M$. What is the definition of compact open $C^{r}$-topology on the space of $\mathcal{C}^{r}$-sections?
4
votes
0answers
46 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
6
votes
1answer
70 views

Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form ...